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\begin{document}
\title{{\huge{}Causal Identification in Social Sciences Using Observational
Data}\\
{\normalsize{}\author{F\i rat Bilgel}
\affil{\footnotesize{}Department of Economics, MEF University, Istanbul 34396 Turkey. email: \texttt{\protect\href{mailto:}{bilgelf@mef.edu.tr}}}}}
\maketitle
\begin{quotation}
\emph{\vspace{1cm}
}
\end{quotation}

\setcounter{figure}{1}

\begin{figure}[H]
\centering{}\caption{\label{fig:racial-bias}Racial bias and police use of force}
\subfloat[\label{fig:barop}Original DAG]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state, ] (2) [right =of 1, xshift=0.1cm, yshift=1.3cm] {$M$};     
\node[state, red] (3) [right =of 1, xshift=2cm] {$Y$};     
\node[state] (4) [below =of 2, xshift=0cm, yshift=-1cm] {$X$};     
\node[state,fill=gray!30!white] (5) [above =of 2, xshift=1.8cm,yshift=-1.4cm] {$U$};  
\node[state, white, double] (6) [above =of 2, xshift=0cm, yshift=-0.2cm] {$S$};      
\path (1) edge node[above] {} (2); 
\path (2) edge node[above] {} (3);    
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\path (4) edge node[el,above] {} (1);
\path (4) edge node[el,above] {} (2);
\path (4) edge node[el,above] {} (3);
\path (5) edge node[above] {} (2); 
\path (5) edge node[above] {} (3); 
%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}}\hspace{0.5cm}\subfloat[\label{fig:selection}Selection mechanism]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state, ] (2) [right =of 1, xshift=0.1cm, yshift=1.3cm] {$M$};     
\node[state, red] (3) [right =of 1, xshift=2cm] {$Y$};     
\node[state] (4) [below =of 2, xshift=0cm, yshift=-1cm] {$X$};     
\node[state,fill=gray!30!white] (5) [above =of 2, xshift=1.8cm,yshift=-1.4cm] {$U$};   
\node[state, double] (6) [above =of 2, xshift=-0cm, yshift=-0.2cm] {$S$};     
\path (1) edge node[above] {} (2); 
\path (2) edge node[above] {} (3);    
\path (1) edge node[el,above] {} (3);
\path (4) edge node[el,above] {} (1);
\path (4) edge node[el,above] {} (2);
\path (4) edge node[el,above] {} (3);
\path (5) edge node[above] {} (2); 
\path (5) edge node[above] {} (3); 
\path (2) edge node[above] {} (6); 
%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}}\hspace{0.5cm}\subfloat[\label{fig:911}911 Calls]{
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\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state, ] (2) [right =of 1, xshift=0.1cm, yshift=1.3cm] {$M$};     
\node[state, red] (3) [right =of 1, xshift=2cm] {$Y$};     
\node[state] (4) [below =of 2, xshift=0cm, yshift=-1cm] {$X_{1}$};     
\node[state,fill=gray!30!white] (5) [above =of 2, xshift=1.8cm,yshift=-1.4cm] {$U$};   
\node[state, double] (6) [above =of 2, xshift=-0cm, yshift=-0.2cm] {$S$};     
\node[state] (7) [right =of 2, xshift=-0.7cm, yshift=1cm] {$C$};  
\node[state] (8) [above =of 1, xshift=0.5cm, yshift=0.5cm] {$X_{2}$};  
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\path (7) edge node[above] {} (3);
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%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}}
\end{figure}

\begin{figure}[H]
	\caption{\label{fig:internal-IV}A DAG with an internal IV}
	
	\centering{}\subfloat[\label{fig:internal-IV-explicit}Unobservables are explicitly shown]{
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		\resizebox{0.36\textwidth}{!}{
			\begin{tikzpicture}
				\node[state] (1) {$X_{1}$};     
				\node[state,red] (2) [right =of 1, xshift=4cm] {$Y$};  
				\node[state,blue] (3) [above =of 1, xshift=2.8cm, yshift=-0.9cm] {$T$};   
				\node[state,magenta] (4) [above =of 3, yshift=0cm] {$Z_{1}$};     
				\node[state] (5) [above=of 4, yshift=0cm] {$X_{2}$};    
				\node[state,fill=gray!30!white] (6) [above =of 1, yshift=1.3cm] {$U_{1}$};            
				\node[state,fill=gray!30!white] (7) [above =of 2, yshift=2cm] {$U_{2}$};             
				\node[state,fill=gray!30!white] (8) [above =of 2, xshift=0cm, yshift=0cm] {$U_{3}$};             
				\node[state,fill=gray!30!white] (9) [below =of 3, xshift=0cm] {$U_{4}$};             
				\node[state] (10) [above=of 1, xshift=0.7cm , yshift=0cm] {$Z_{2}$}; 
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				\path (3) edge node[below] {} (2);    
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				\path (1) edge node[el,below] {} (2); 
				\path (4) edge node[el,above] {} (3); 
				\path (4) edge node[below] {} (10);
				\path (10) edge node[below] {} (1);
				\path (5) edge[bend left=40] node[below] {} (3);
				\path (6) edge node[above] {} (1); 
				\path (6) edge node[above] {} (3); 
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				\path (7) edge node[above] {} (5);
				\path (7) edge node[above] {} (3);
				\path (7) edge node[above] {} (4);
				\path (8) edge node[above] {} (5);
				\path (8) edge node[above] {} (3);
				\path (8) edge node[above] {} (2);
				\path (9) edge node[above] {} (1);
				\path (9) edge node[above] {} (2);
				%\path[bidirected] (10) edge[bend left=20] node[el,below] {\footnotesize{}} (5);
			\end{tikzpicture}
	}}\hspace{1cm}\subfloat[\label{fig:marginternal-IV}Marginally equivalent graph $\mathbf{G_{M}}$]{
		\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.8 cm},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
		\resizebox{0.36\textwidth}{!}{
			\begin{tikzpicture}
				\node[state] (1) {$X_{1}$};     
				\node[state,red] (2) [right =of 1, xshift=4cm] {$Y$};  
				\node[state,blue] (3) [above =of 1, xshift=2.8cm, yshift=-0.9cm] {$T$};   
				\node[state,magenta] (4) [above =of 3, yshift=0cm] {$Z_{1}$};     
				\node[state] (5) [above=of 4, yshift=0cm] {$X_{2}$};               
				\node[state] (10) [above=of 1, xshift=0.7cm , yshift=0cm] {$Z_{2}$}; 
				\path[bidirected] (1) edge[bend right=45] node[el,above] {\footnotesize{}} (2);
				\path[bidirected] (1) edge[bend left=45] node[el,above] {\footnotesize{}} (5);
				\path (1) edge node[above] {} (3); 
				\path (3) edge node[below] {} (2);    
				%\path (4) edge node[el,below] {} (5); 
				\path (1) edge node[el,below] {} (2);
				\path (4) edge node[el,above] {} (3); 
				\path[bidirected] (2) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
				\path[bidirected] (4) edge[bend left=45] node[el,above] {\footnotesize{}} (3);
				\path[bidirected] (5) edge[bend right=45] node[el,below] {\footnotesize{}} (3);
				\path[bidirected] (4) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
				\path[bidirected] (3) edge[bend left=45] node[el,above] {\footnotesize{}} (2);
				\path[bidirected] (3) edge[bend right=45] node[el,above] {\footnotesize{}} (1);
				\path (4) edge node[below] {} (10);
				\path (10) edge node[below] {} (1);
				\path (5) edge[bend left=60] node[below] {} (3);
				%\path[bidirected] (10) edge[bend left=20] node[el,below] {\footnotesize{}} (5);
			\end{tikzpicture}
	}}
\end{figure}	


\begin{figure}[H]
\caption{\label{fig:valid-invalid}Mutilated graphs $\mathbf{G_{\underline{T}}}$}

\centering{}\subfloat[\label{fig:valid-1}Adjusted for $Z_{2},X_{1}$ (admissible IV)]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.8 cm},   square_state/.style ={rectangle, draw, minimum width = 0.8 cm},  point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.4\textwidth}{!}{
\begin{tikzpicture}
\node[state,square_state] (1) {$X_{1}$};     
\node[state,red] (2) [right =of 1, xshift=4cm] {$Y$};  
\node[state,blue] (3) [above =of 1, xshift=2.8cm, yshift=-0.9cm] {$T$};   
\node[state,magenta] (4) [above =of 3, yshift=0cm] {$Z_{1}$};     
\node[state] (5) [above=of 4, yshift=0cm] {$X_{2}$};               
\node[state, square_state] (10) [above=of 1, xshift=0.7cm , yshift=0cm] {$Z_{2}$}; 
\path[bidirected] (1) edge[bend right=45] node[el,above] {\footnotesize{}} (2);
\path[bidirected] (1) edge[bend left=45] node[el,above] {\footnotesize{}} (5);
\path (1) edge node[above] {} (3); 
%\path (3) edge node[below] {} (2);    
%\path[red] (4) edge node[el,below] {} (5); 
\path (1) edge node[el,below] {} (2);
\path (4) edge node[el,above] {} (3); 
\path[bidirected] (2) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
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\path[bidirected] (5) edge[bend right=45] node[el,below] {\footnotesize{}} (3);
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\path[bidirected] (3) edge[bend left=45] node[el,above] {\footnotesize{}} (2);
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\path (4) edge node[below] {} (10);
\path (10) edge node[below] {} (1);
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%\path[bidirected] (10) edge[bend left=20] node[el,below] {\footnotesize{}} (5);
\end{tikzpicture}
}}\hspace{1cm}\subfloat[\label{fig:invalid}Adjusted for $Z_{2},X_{1},X_{2}$ (inadmissible
IV)]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.8 cm},   square_state/.style ={rectangle, draw, minimum width = 0.8 cm},  point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.4\textwidth}{!}{
\begin{tikzpicture}
\node[state,square_state] (1) {$X_{1}$};     
\node[state,red] (2) [right =of 1, xshift=4cm] {$Y$};  
\node[state,blue] (3) [above =of 1, xshift=2.8cm, yshift=-0.9cm] {$T$};   
\node[state,magenta] (4) [above =of 3, yshift=0cm] {$Z_{1}$};     
\node[state, square_state] (5) [above=of 4, yshift=0cm] {$X_{2}$};               
\node[state, square_state] (10) [above=of 1, xshift=0.7cm , yshift=0cm] {$Z_{2}$}; 
\path[bidirected] (1) edge[bend right=45] node[el,above] {\footnotesize{}} (2);
\path[bidirected] (1) edge[bend left=45] node[el,above] {\footnotesize{}} (5);
\path (1) edge node[above] {} (3); 
%\path (3) edge node[below] {} (2);    
%\path[red] (4) edge node[el,below] {} (5); 
\path (1) edge node[el,below] {} (2);
\path (4) edge node[el,above] {} (3); 
\path[bidirected, red] (2) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
\path[bidirected] (4) edge[bend left=45] node[el,above] {\footnotesize{}} (3);
\path[bidirected] (5) edge[bend right=45] node[el,below] {\footnotesize{}} (3);
\path[bidirected, red] (4) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
\path[bidirected] (3) edge[bend left=45] node[el,above] {\footnotesize{}} (2);
\path[bidirected] (3) edge[bend right=45] node[el,above] {\footnotesize{}} (1);
\path (4) edge node[below] {} (10);
\path (10) edge node[below] {} (1);
\path (5) edge[bend left=60] node[below] {} (3);
%\path[bidirected] (10) edge[bend left=20] node[el,below] {\footnotesize{}} (5);
\end{tikzpicture}
}}
\end{figure}

\begin{figure}[H]
	\caption{\label{fig:frontdoor}Rigour of defense strategy as a treatment-outcome
		mediator}
	
	\centering{}\subfloat[\label{fig:frontdoor-explicit}Unobservables are explicitly shown]{
		\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.8 cm},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
		\resizebox{0.36\textwidth}{!}{
			\begin{tikzpicture}
				\node[state] (1) {$X_{1}$};     
				\node[state,red] (2) [right =of 1, xshift=4cm] {$Y$};  
				\node[state,blue] (3) [above =of 1, xshift=2.8cm, yshift=-0.9cm] {$T$};   
				\node[state,magenta] (4) [above =of 3, yshift=0cm] {$Z_{1}$};     
				\node[state] (5) [above=of 4, yshift=0cm] {$X_{2}$};    
				\node[state,fill=gray!30!white] (6) [above =of 1, yshift=1.3cm] {$U_{1}$};            
				\node[state,fill=gray!30!white] (7) [above =of 2, yshift=2cm] {$U_{2}$};             
				\node[state,fill=gray!30!white] (8) [above =of 2, xshift=0cm, yshift=0cm] {$U_{3}$};             
				\node[state,fill=gray!30!white] (9) [below =of 3, xshift=0cm] {$U_{4}$};             
				\node[state] (10) [above=of 1, xshift=0.7cm , yshift=0cm] {$Z_{2}$}; 
				\node[state] (11) [right=of 3, xshift=-0.3cm,yshift=-0.4cm] {$Q$};    
				\path (1) edge node[above] {} (3); 
				%\path (3) edge node[below] {} (2);    
				\path (3) edge node[below] {} (11);    
				\path (11) edge node[below] {} (2);    
				%\path (4) edge node[el,below] {} (5); 
				\path (1) edge node[el,below] {} (2); 
				\path (4) edge node[el,above] {} (3); 
				\path (4) edge node[below] {} (10);
				\path (10) edge node[below] {} (1);
				\path (5) edge[bend left=40] node[below] {} (3);
				\path (6) edge node[above] {} (1); 
				\path (6) edge node[above] {} (3); 
				\path (6) edge node[above] {} (5); 
				\path (7) edge node[above] {} (5);
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				%\path[bidirected] (10) edge[bend left=20] node[el,below] {\footnotesize{}} (5);
			\end{tikzpicture}
	}}\hspace{1cm}\subfloat[\label{fig:frontdoor-Marginally-equivalent-graph}Marginally equivalent
	graph]{
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		\resizebox{0.4\textwidth}{!}{
			\begin{tikzpicture}
				\node[state] (1) {$X_{1}$};     
				\node[state,red] (2) [right =of 1, xshift=4cm] {$Y$};  
				\node[state,blue] (3) [above =of 1, xshift=2.8cm, yshift=-0.9cm] {$T$};   
				\node[state,magenta] (4) [above =of 3, yshift=0cm] {$Z_{1}$};     
				\node[state] (5) [above=of 4, yshift=0cm] {$X_{2}$};               
				\node[state] (6) [right=of 3, xshift=-0.3cm,yshift=-0.4cm] {$Q$};               
				\node[state] (10) [above=of 1, xshift=0.7cm , yshift=0cm] {$Z_{2}$}; 
				\path[bidirected] (1) edge[bend right=45] node[el,above] {\footnotesize{}} (2);
				\path[bidirected] (1) edge[bend left=45] node[el,above] {\footnotesize{}} (5);
				\path (1) edge node[above] {} (3); 
				\path (3) edge node[below] {} (6);    
				\path (6) edge node[below] {} (2);
				%\path (4) edge node[el,below] {} (5); 
				\path (1) edge node[el,below] {} (2);
				\path (4) edge node[el,above] {} (3); 
				\path[bidirected] (2) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
				\path[bidirected] (4) edge[bend left=45] node[el,above] {\footnotesize{}} (3);
				\path[bidirected] (5) edge[bend right=45] node[el,below] {\footnotesize{}} (3);
				\path[bidirected] (4) edge[bend right=45] node[el,above] {\footnotesize{}} (5);
				\path[bidirected] (3) edge[bend left=45] node[el,above] {\footnotesize{}} (2);
				\path[bidirected] (3) edge[bend right=45] node[el,above] {\footnotesize{}} (1);
				\path (4) edge node[below] {} (10);
				\path (10) edge node[below] {} (1);
				\path (5) edge[bend left=60] node[below] {} (3);
				%\path[bidirected] (10) edge[bend left=20] node[el,below] {\footnotesize{}} (5);
			\end{tikzpicture}
	}}
\end{figure}


\begin{center}
\begin{figure}[H]
\caption{\label{fig:reform}Judicial reforms and sentencing disparities }

\centering{}\subfloat[\label{fig:judicial-original}Causal graph $\mathbf{G}$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.8 cm},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.35\textwidth}{!}{
\begin{tikzpicture}
\node[state, blue] (1) {$M$};     
\node[state,fill=gray!30!white] (2) [right =of 1, xshift=0.8cm,yshift=-0cm] {$W$}; 
\node[state, blue] (3)  [right=of 2, xshift=0.8cm, yshift=0cm] {$J$};     
\node[state, ] (4) [below=of 3, xshift=0cm, yshift=-2cm] {$P$};     
\node[state, ] (5) [below =of 1, xshift=0cm, yshift=-2cm] {$Z$};    
\node[state, red] (6) [below =of 2, xshift=1.4cm, yshift=0.5cm] {$Y$};     
\node[state,fill=gray!30!white] (7) [below =of 2, xshift=0cm,yshift=-1cm] {$U$};  
\path (2) edge node[above] {} (3); 
\path (2) edge node[above] {} (1);    
\path (2) edge node[el,above] {} (5);
\path (1) edge node[el,above] {} (6);
\path (1) edge node[above] {} (5); 
\path (5) edge node[el,above] {} (4);
\path (4) edge node[el,above] {} (3);
\path (7) edge node[el,above] {} (4);
\path (7) edge node[el,above] {} (6);
\path (3) edge node[el,above] {} (6);
%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}
}}\hspace{1cm}\subfloat[\label{fig:judicial-equivalent}Marginally equivalent graph $\mathbf{G_{M}}$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.8 cm},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.35\textwidth}{!}{
\begin{tikzpicture}
\node[state, blue] (1) {$M$};     
%\node[state,fill=gray!30!white] (2) [right =of 1, xshift=0.8cm,yshift=-0cm] {$W$}; 
\node[state, blue] (3)  [right=of 1, xshift=3.3cm, yshift=0cm] {$J$};     
\node[state, ] (4) [below=of 3, xshift=0cm, yshift=-2cm] {$P$};     
\node[state, ] (5) [below =of 1, xshift=0cm, yshift=-2cm] {$Z$};    
\node[state, red] (6) [below =of 1, xshift=3.4cm, yshift=0.5cm] {$Y$};     
%\node[state,fill=gray!30!white] (7) [below =of 2, xshift=0cm,yshift=-1cm] {$U$};  
\path (1) edge node[el,above] {} (6);
\path (1) edge node[above] {} (5); 
\path (5) edge node[el,above] {} (4);
\path (4) edge node[el,above] {} (3);
\path (3) edge node[el,above] {} (6);
\path[bidirected] (1) edge[bend left=0] node[el,above] {} (3);
\path[bidirected] (1) edge[bend left=40] node[el,above] {} (5);
\path[bidirected] (4) edge[bend left=40] node[el,above] {} (6);
\path[bidirected] (3) edge[bend right=35] node[el,above] {} (5);
\end{tikzpicture}
}}
\end{figure}
\par\end{center}


\begin{center}
\begin{figure}[H]
\caption{\label{fig:coverage}Media coverage and Sentencing}

\centering{}\subfloat[\label{fig:Causally-independent-mediators}Causally independent mediators]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state, ] (2) [right =of 1, xshift=0.8cm, yshift=2.5cm] {$M$};     
\node[state, ] (3) [right =of 1, xshift=0.8cm, yshift=0.9cm] {$B$};    
\node[state, red] (4) [right =of 1, xshift=3.2cm] {$Y$};     
\node[state] (5) [below =of 3, xshift=0cm, yshift=-0.7cm] {$X$};     
\path (1) edge node[above] {$\beta$} (2); 
%\path (2) edge node[below, pos=0.3, xshift=-0.3cm]  {$\lambda$} (3);    
\path (1) edge node[el,above] {$\gamma$} (3);
\path (1) edge node[el,above, xshift=-0.5cm] {$\alpha$} (4);
\path (2) edge node[above] {$\delta$} (4);
\path (3) edge node[el,above] {$\tau$} (4);
\path (5) edge node[above] {} (1); 
\path (5) edge node[above] {} (4); 
\path (5) edge[bend right=35] node[above] {} (2); 
\path (5) edge node[above] {} (3); 
%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}}\hspace{1cm}\subfloat[\label{fig:Causally-dependent-mediators}Causally dependent mediators]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state, ] (2) [right =of 1, xshift=0.8cm, yshift=2.5cm] {$M$};     
\node[state, ] (3) [right =of 1, xshift=0.8cm, yshift=0.9cm] {$B$};    
\node[state, red] (4) [right =of 1, xshift=3.2cm] {$Y$};     
\node[state] (5) [below =of 3, xshift=0cm, yshift=-0.7cm] {$X$};     
\path (1) edge node[above] {$\beta$} (2); 
\path (2) edge node[below, pos=0.3, xshift=-0.3cm]  {$\lambda$} (3);    
\path (1) edge node[el,above] {$\gamma$} (3);
\path (1) edge node[el,above, xshift=-0.5cm] {$\alpha$} (4);
\path (2) edge node[above] {$\delta$} (4);
\path (3) edge node[el,above] {$\tau$} (4);
\path (5) edge node[above] {} (1); 
\path (5) edge node[above] {} (4); 
\path (5) edge[bend right=35] node[above] {} (2); 
\path (5) edge node[above] {} (3); 
%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}}
\end{figure}
\par\end{center}


\begin{center}
\begin{figure}[H]
\caption{\label{fig:CIM-1}Direct and indirect effects with causally independent
mediators}

\centering{}\subfloat[\label{fig:beta-1}Effect of $T$ on $M$, adjusted for $X$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},  square_state/.style ={rectangle, draw, minimum width = 0.8 cm},   point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.2\textwidth}{!}{
\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state, red] (2) [right =of 1, xshift=0.1cm, yshift=2.3cm] {$M$};     
\node[state] (3) [right =of 1, xshift=0.1cm, yshift=0.9cm] {$B$};    
\node[state] (4) [right =of 1, xshift=2cm] {$Y$};     
\node[state, square_state] (5) [below =of 3, xshift=0cm, yshift=-0.7cm] {$X$};     
\path[cadmiumgreen] (1) edge node[above] {$\beta$} (2); 
%\path (2) edge node[below, pos=0.3, xshift=-0.3cm]  {$\lambda$} (3);  
\path (1) edge node[el,above] {$\gamma$} (3);
\path (1) edge node[el,above, xshift=-0.5cm] {$\alpha$} (4);
\path (2) edge node[above] {$\delta$} (4);
\path (3) edge node[el,above] {$\tau$} (4);
\path (5) edge node[above] {} (1); 
\path (5) edge node[above] {} (4); 
\path (5) edge[bend right=35] node[above] {} (2); 
\path (5) edge node[above] {} (3); 
\end{tikzpicture}
}}\hspace{0.5cm}\subfloat[\label{fig:delta}Effect of $M$ on $Y$, adjusted for $T,X$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},   square_state/.style ={rectangle, draw, minimum width = 0.8 cm},  point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.2\textwidth}{!}{
\begin{tikzpicture}        
\node[state, square_state] (1) {$T$};     
\node[state, blue] (2) [right =of 1, xshift=0.1cm, yshift=2.3cm] {$M$};     
\node[state] (3) [right =of 1, xshift=0.1cm, yshift=0.9cm] {$B$};    
\node[state, red] (4) [right =of 1, xshift=2cm] {$Y$};     
\node[state, square_state] (5) [below =of 3, xshift=0cm, yshift=-0.7cm] {$X$};     
\path (1) edge node[above] {$\beta$} (2); 
%\path[cadmiumgreen] (2) edge node[below, pos=0.3, xshift=-0.3cm]  {$\lambda$} (3);  
\path (1) edge node[el,above] {$\gamma$} (3);
\path (1) edge node[el,above, xshift=-0.5cm] {$\alpha$} (4);
\path[cadmiumgreen] (2) edge node[above] {$\delta$} (4);
\path (3) edge node[el,above] {$\tau$} (4);
\path (5) edge node[above] {} (1); 
\path (5) edge node[above] {} (4); 
\path (5) edge[bend right=35] node[above] {} (2); 
\path (5) edge node[above] {} (3); 
%\path[bidirected] (2) edge[bend left=40] node[el,above, rotate=30] {\footnotesize{$U$}} (3);
\end{tikzpicture}
}}\hspace{0.5cm}\subfloat[\label{fig:gamma}Effect of $T$ on $B$, adjusted for $X$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},  square_state/.style ={rectangle, draw, minimum width = 0.8 cm},   point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.2\textwidth}{!}{
\begin{tikzpicture}        
\node[state, blue] (1) {$T$};     
\node[state] (2) [right =of 1, xshift=0.1cm, yshift=2.3cm] {$M$};     
\node[state, red] (3) [right =of 1, xshift=0.1cm, yshift=0.9cm] {$B$};    
\node[state] (4) [right =of 1, xshift=2cm] {$Y$};     
\node[state, square_state] (5) [below =of 3, xshift=0cm, yshift=-0.7cm] {$X$};     
\path (1) edge node[above] {$\beta$} (2); 
%\path (2) edge node[below, pos=0.3, xshift=-0.3cm]  {$\lambda$} (3);  
\path[cadmiumgreen] (1) edge node[el,above] {$\gamma$} (3);
\path (1) edge node[el,above, xshift=-0.5cm] {$\alpha$} (4);
\path (2) edge node[above] {$\delta$} (4);
\path (3) edge node[el,above] {$\tau$} (4);
\path (5) edge node[above] {} (1); 
\path (5) edge node[above] {} (4); 
\path (5) edge[bend right=35] node[above] {} (2); 
\path (5) edge node[above] {} (3); 
\end{tikzpicture}
}}\hspace{0.5cm}\subfloat[\label{fig:tau-1}Effect of $B$ on $Y$, adjusted for $T,X$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},   square_state/.style ={rectangle, draw, minimum width = 0.8 cm},  point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.2\textwidth}{!}{
\begin{tikzpicture}        
\node[state, square_state] (1) {$T$};     
\node[state, square_state] (2) [right =of 1, xshift=0.1cm, yshift=2.3cm] {$M$};     
\node[state, blue] (3) [right =of 1, xshift=0.1cm, yshift=0.9cm] {$B$};    
\node[state, red] (4) [right =of 1, xshift=2cm] {$Y$};     
\node[state, square_state] (5) [below =of 3, xshift=0cm, yshift=-0.7cm] {$X$};     
\path (1) edge node[above] {$\beta$} (2); 
%\path (2) edge node[below, pos=0.3, xshift=-0.3cm]  {$\lambda$} (3);  
\path (1) edge node[el,above] {$\gamma$} (3);
\path (1) edge node[el,above, xshift=-0.5cm] {$\alpha$} (4);
\path (2) edge node[above] {$\delta$} (4);
\path[cadmiumgreen] (3) edge node[el,above] {$\tau$} (4);
\path (5) edge node[above] {} (1); 
\path (5) edge node[above] {} (4); 
\path (5) edge[bend right=35] node[above] {} (2); 
\path (5) edge node[above] {} (3); 
\end{tikzpicture}
}}
\end{figure}
\par\end{center}

\begin{figure}[H]
\caption{\label{fig:recid-pc}Causal discovery - edge orientations (PC algorithm)}

\begin{centering}
\subfloat[\label{fig:CPDAG-via-PC}CPDAG]{
\centering{}\includegraphics[width=0.35\textwidth]{pc_undirected}}\subfloat[\label{fig:DAG-PC}Corresponding DAG]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.33\textwidth}{!}{
\begin{tikzpicture}        
\node[state] (1) {$X_{1}$};     
\node[state, red] (2) [right =of 1, xshift=0cm, yshift=0cm] {$Y$};     
\node[state] (3) [right =of 2, xshift=0.8cm, yshift=0cm] {$G$};     
\node[state] (4) [above =of 3, xshift=0cm, yshift=-0.9cm] {$S$};     
\node[state] (5) [above =of 4, xshift=0cm, yshift=-0.9cm] {$M$};     
\node[state] (6) [below=of 3, xshift=0cm, yshift=0.9cm] {$A$};     
\node[state] (7) [right =of 2, xshift=2.2cm, yshift=0cm] {$P$};     
\node[state] (8) [above =of 1, xshift=0cm, yshift=0.8cm] {$X_{2}$};     
\node[state] (9) [below =of 1, xshift=0.2cm, yshift=0.2cm] {$F$};     
\node[state] (10) [above =of 2, xshift=-0.8cm, yshift=0cm] {$R$};     
\node[state] (11) [above =of 2, xshift=0.8cm, yshift=0cm] {$D$};     
\node[state] (12) [above =of 11, xshift=0.8cm, yshift=0cm] {$B$};     
\node[state, blue] (13) [below=of 12, xshift=0cm, yshift=-4cm] {$T$};     
\path (9) edge[bend left=30] node[el,above] {} (8);
\path (8) edge[bend left=25] node[el,above] {} (12);
\path (7) edge[bend right=35] node[el,above] {} (12);
\path (7) edge[bend left=30] node[el,above] {} (13);
\path (9) edge[bend right=25] node[el,above] {} (13);
\path (1) edge node[above] {} (13); 
\path (8) edge[bend right=25] node[above] {} (2); 
\path (1) edge node[above] {} (8); 
\path (10) edge node[above] {} (2); 
\path (11) edge node[above] {} (2); 
\path (11) edge node[above] {} (12); 
\path (12) edge node[above] {} (13); 
\path (13) edge node[above] {} (2); 
\path (9) edge node[above] {} (2); 
\path (3) edge node[above] {} (2); 
\path (4) edge node[above] {} (2); 
\path (5) edge node[above] {} (2); 
\path (6) edge node[above] {} (7); 
\path (4) edge node[above] {} (7); 
\path (3) edge node[above] {} (7); 
\path (5) edge node[above] {} (12); 
\end{tikzpicture}
}}
\par\end{centering}
\end{figure}


\begin{figure}[H]
\caption{\label{fig:recid-rfci}Causal discovery - edge orientations (RFCI
or GFCI algorithm)}

\begin{centering}
\subfloat[\label{fig:PAG-via-RFCI}PAG]{
\centering{}\includegraphics[width=0.33\textwidth]{rfci}}\subfloat[\label{fig:DAG-RFCI-1}Corresponding DAG$^{\dagger}$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.3\textwidth}{!}{
\begin{tikzpicture}        
\node[state] (1) {$X_{1}$};     
\node[state, red] (2) [right =of 1, xshift=0cm, yshift=0cm] {$Y$};     
\node[state] (3) [right =of 2, xshift=0.8cm, yshift=0cm] {$G$};     
\node[state] (4) [above =of 3, xshift=0cm, yshift=-0.9cm] {$S$};     
\node[state] (5) [above =of 4, xshift=0cm, yshift=-0.9cm] {$M$};     
\node[state] (6) [below=of 3, xshift=0cm, yshift=0.9cm] {$A$};     
\node[state] (7) [right =of 2, xshift=2.2cm, yshift=0cm] {$P$};     
\node[state] (8) [above =of 1, xshift=0cm, yshift=0.8cm] {$X_{2}$};     
\node[state] (9) [below =of 1, xshift=0.2cm, yshift=0.2cm] {$F$};     
\node[state] (10) [above =of 2, xshift=-0.8cm, yshift=0cm] {$R$};     
\node[state] (11) [above =of 2, xshift=0.8cm, yshift=0cm] {$D$};     
\node[state] (12) [above =of 11, xshift=0.8cm, yshift=0cm] {$B$};     
\node[state, blue] (13) [below=of 12, xshift=0cm, yshift=-4cm] {$T$};     
\path (9) edge[bend left=30] node[el,above] {} (8);
\path (8) edge[bend left=25] node[el,above] {} (12);
\path (8) edge[bend right=25] node[above] {} (2); 
\path (7) edge[bend right=35] node[el,above] {} (12);
\path (7) edge[bend left=30] node[el,above] {} (13);
\path (9) edge[bend right=25] node[el,above] {} (13);
\path (1) edge node[above] {} (13); 
\path (1) edge node[above] {} (8); 
\path (10) edge node[above] {} (2); 
\path (11) edge node[above] {} (2); 
\path (11) edge node[above] {} (12); 
\path (12) edge node[above] {} (13); 
\path (13) edge node[above] {} (2); 
\path (9) edge node[above] {} (2); 
\path (3) edge node[above] {} (2); 
\path (4) edge node[above] {} (2); 
\path (5) edge node[above] {} (2); 
\path (6) edge node[above] {} (7); 
\path (4) edge node[above] {} (7); 
\path (3) edge node[above] {} (7); 
\path (5) edge node[above] {} (12); 
\path[bidirected] (1) edge[bend left=20] node[el,above] {} (8);
\path[bidirected] (8) edge[bend left=30] node[el,above] {} (9);
\path[bidirected] (10) edge[bend left=30] node[el,above] {} (2);
%\path[bidirected] (11) edge[bend left=30] node[el,above] {} (2);
\path[bidirected] (11) edge[bend left=30] node[el,above] {} (12);
%\path[bidirected] (1) edge[bend left=30] node[el,above] {} (13);
%\path[bidirected] (5) edge[bend right=30] node[el,above] {} (2);
\path[bidirected] (5) edge[bend right=30] node[el,above] {} (12);
\path[bidirected] (4) edge[bend left=30] node[el,above] {} (7);
\path[bidirected] (3) edge[bend right=30] node[el,above] {} (7);
\path[bidirected] (6) edge[bend right=30] node[el,above] {} (7);
%\path[bidirected] (9) edge[bend left=30] node[el,above] {} (13);
\end{tikzpicture}
}}\subfloat[\label{fig:DAG-RFCI-2}Corresponding DAG$^{\ddagger}$]{
\centering{}\tikzset{     -Latex,auto,node distance =1 cm and 1 cm,semithick ,     state/.style ={circle, draw, minimum width = 0.7 cm, font=\scriptsize},     point/.style = {circle, draw, inner sep=0.01cm,fill,node contents={}},     bidirected/.style={Latex-Latex,dashed},     el/.style = {inner sep=2pt, align=left, sloped} }
\resizebox{0.30\textwidth}{!}{
\begin{tikzpicture}        
\node[state] (1) {$X_{1}$};     
\node[state, red] (2) [right =of 1, xshift=0cm, yshift=0cm] {$Y$};     
\node[state] (3) [right =of 2, xshift=0.8cm, yshift=0cm] {$G$};     
\node[state] (4) [above =of 3, xshift=0cm, yshift=-0.9cm] {$S$};     
\node[state] (5) [above =of 4, xshift=0cm, yshift=-0.9cm] {$M$};     
\node[state, magenta] (6) [below=of 3, xshift=0cm, yshift=0.9cm] {$A$};     
\node[state] (7) [right =of 2, xshift=2.2cm, yshift=0cm] {$P$};     
\node[state] (8) [above =of 1, xshift=0cm, yshift=0.8cm] {$X_{2}$};     
\node[state] (9) [below =of 1, xshift=0.2cm, yshift=0.2cm] {$F$};     
\node[state] (10) [above =of 2, xshift=-0.8cm, yshift=0cm] {$R$};     
\node[state] (11) [above =of 2, xshift=0.8cm, yshift=0cm] {$D$};     
\node[state] (12) [above =of 11, xshift=0.8cm, yshift=0cm] {$B$};     
\node[state, blue] (13) [below=of 12, xshift=0cm, yshift=-4cm] {$T$};     
\path (9) edge[bend left=30] node[el,above] {} (8);
\path (8) edge[bend right=25] node[above] {} (2); 
\path (8) edge[bend left=25] node[el,above] {} (12);
\path (7) edge[bend right=35] node[el,above] {} (12);
\path (7) edge[bend left=30] node[el,above] {} (13);
\path (9) edge[bend right=25] node[el,above] {} (13);
\path (1) edge node[above] {} (13); 
\path (1) edge node[above] {} (8); 
\path (10) edge node[above] {} (2); 
\path (11) edge node[above] {} (2); 
\path (11) edge node[above] {} (12); 
\path (12) edge node[above] {} (13); 
\path (13) edge node[above] {} (2); 
\path (9) edge node[above] {} (2); 
\path (3) edge node[above] {} (2); 
\path (4) edge node[above] {} (2); 
\path (5) edge node[above] {} (2); 
\path (6) edge node[above] {} (7); 
\path (4) edge node[above] {} (7); 
\path (3) edge node[above] {} (7); 
\path (5) edge node[above] {} (12); 
\path[bidirected] (9) edge[bend right=30] node[el,above] {} (8);
\path[bidirected] (9) edge[bend left=25] node[el,above] {} (13);
\path[bidirected] (1) edge[bend right=20] node[above] {} (13); 
\path[bidirected] (1) edge[bend left=20] node[above] {} (8); 
\path[bidirected] (10) edge[bend left=30] node[above] {} (2); 
\path[bidirected] (11) edge[bend right=30] node[above] {} (2); 
\path[bidirected] (11) edge[bend left=30] node[above] {} (12); 
\path[bidirected] (9) edge[bend right=30] node[above] {} (2); 
\path[bidirected] (3) edge[bend left=30] node[above] {} (2); 
\path[bidirected] (4) edge[bend right=30] node[above] {} (2); 
\path[bidirected] (5) edge[bend right=25] node[above] {} (2); 
\path[bidirected] (6) edge[bend right=30] node[above] {} (7); 
\path[bidirected] (4) edge[bend left=30] node[above] {} (7); 
\path[bidirected] (3) edge[bend right=30] node[above] {} (7); 
\path[bidirected] (5) edge[bend right=30] node[above] {} (12); 
\end{tikzpicture}
}}
\par\end{centering}
\end{figure}

\end{document}
